President Obama’s re-election and Democratic gains in Congress were supposed to make it easier for the party to strike a deal with Republicans to resolve the year-end fiscal crisis by providing new leverage. But they could also make it harder as empowered Democrats, including some elected on liberal platforms, resist significant changes in entitlement programs like Social Security and Medicare.

As Congress returned Monday, the debate over those programs, which many Democrats see as the core of the party’s identity, was shaping up as the Democratic version of the higher-profile struggle among Republicans over taxes.

In failed deficit reduction talks last year, Mr. Obama signaled a willingness to consider substantial changes in the social safety net, including a gradual increase in the eligibility age for Medicare and limits in the growth rate of future Social Security benefits. An urgent question hanging over the new round of deficit talks is which of those changes Mr. Obama and Congressional Democrats would accept today....